Derman, Implied and local volatility, etc

grantx · Nov 13, 2006

Perhaps someone can clarify a few points for me re Emanuel Derman’s paper, “Modeling the Volatility Smile” ,October 2006 http://finmath.stanford.edu/seminars/documents/Stanford.Smile.Derman.pdf

Generally, how is the volatility of volatility determined (I can determine implied by iteration)?

How do I reconcile these statements (page 6):”Volatility of Implied Volatility Decreases with Expiration” and “Short-term implied volatilities are more volatile”? Presumably, (short-term, reducing ) volatility of implied is derived from (short-term, more volatile) implieds?

Continuing from above: “This suggests mean reversion or stationarity …”. ”Stationarity” is not in my English dictionary (probably in an American). Does this mean stationary? If so, which is it – mean reverting or stationary? Or have quants embraced aspect of quantum physics, now?

Continuing from the points above (page 13): “as assets approach liabilities, equity volatility increases”. I’m assuming here “assets” could refer to the underlying and “liabilities” refers to strikes. Again, this appears to conflict with ”Volatility of Implied Volatility Decreases with Expiration” assuming underlying volatility necessarily transfers to its derivatives.

The Quantitative Strategies Research Notes he jointly produced while at Goldman Sachs are models of clarity and lucidity for the mathematically challenged (myself). Don’t know what happened with this one.

Grant.

Robertral · Dec 4, 2006

grantx said:
Perhaps someone can clarify a few points for me re Emanuel Derman’s paper, “Modeling the Volatility Smile” ,October 2006 http://finmath.stanford.edu/seminars/documents/Stanford.Smile.Derman.pdf

Generally, how is the volatility of volatility determined (I can determine implied by iteration)?

How do I reconcile these statements (page 6):”Volatility of Implied Volatility Decreases with Expiration” and “Short-term implied volatilities are more volatile”? Presumably, (short-term, reducing ) volatility of implied is derived from (short-term, more volatile) implieds?

Continuing from above: “This suggests mean reversion or stationarity …”. ”Stationarity” is not in my English dictionary (probably in an American). Does this mean stationary? If so, which is it – mean reverting or stationary? Or have quants embraced aspect of quantum physics, now?

Continuing from the points above (page 13): “as assets approach liabilities, equity volatility increases”. I’m assuming here “assets” could refer to the underlying and “liabilities” refers to strikes. Again, this appears to conflict with ”Volatility of Implied Volatility Decreases with Expiration” assuming underlying volatility necessarily transfers to its derivatives.

The Quantitative Strategies Research Notes he jointly produced while at Goldman Sachs are models of clarity and lucidity for the mathematically challenged (myself). Don’t know what happened with this one.

Grant.

One way is that once you have the implied vol (from what ever method you choose) as a time series you can calculate the volatility of that time series that will give you the vol of vol.
Mean reverting means that the process will have a long term mean and that if the process moves away from the mean it will revert back to the mean with a mean reversion speed of kappa. i.e vol is a mean reverting process. say its long term mean of 12%, if the vol is different to this then the vol will revert back to its mean of 12% at a speed kappa.

Stikes are not defined as liabilities.

It's been a long time since I have read that paper....if you need any more help message me and I will take the time to read it again.

grantx · Dec 4, 2006

Robert,

Thank you for the info. Useful stuff and really appreciated.

Strange thing is, your suggestion for deriving vol of vol from a time-series is what I did some time ago. However, this was before I came the term "vol of vol".

I am familiar with mean reversion but kappa is a new one. I've always imagined that the further implied moves from the mean, the greater the pull is (similar to "pull-to-par"?) to bring it back - almost like an elastic band affect.

Can kappa be quantified or is it just intuitive, eg kappa will be greater for implied the further the implied is from the mean, that is, the speed of the return will be that more rapid (is that a tautology)?

"Strikes" and "liabilities". So what's being referred to here?

Thanks, once again.

Grant.

Silent.Trader · Dec 5, 2006

grantx said:
"Strikes" and "liabilities". So what's being referred to here?

I did not read the report, but at first glance my interpretation would be that asset is the option and liability expiration. So if the option approaches expiration the volatility of the option price (equity volatility) increases. Consequently this volatility is refering to the volatility of the optionprice itself and not to the implied volatility used for optionpricing.

Grtnx
Wilco

grantx · Dec 7, 2006

Wilco,

Apologies for the delay in response.

From my own observations, as options approach expiration the implied volatility becomes more volatile. This is especially true of deep in-the-money calls. However, this is not necessarily the case with underlying.

This is not surprising really given the minimal - and reducing - time value element, by which the implied is derived/is a reflection of (which?), eg an option with no time value (at a discount) has zero implied volatility.

To continue my blind speculation, I think my definition of strikes and liabilities is still correct.

Assets = underlying, Strikes = liabilities. An option holder has no liability (obligation); the obligation is on the seller, to take delivery (put option) of the underlying at the strike, or make delivery (call option) at the strike. It is a liability because the risk is (theoretically) open-ended - that the value/price of the stock he takes (put option) is less than the strike value; or the value/price of the underlying he delivers (call option) is above the strike value. So one may view the exercise of an option as an exchange - of the cancelling of liabilities (no more obligation) in exchange for an asset (no more asset).

I'm open to suggestions or corrections.

Grant.

Robertral · Dec 7, 2006

grantx said:
Robert,

Thank you for the info. Useful stuff and really appreciated.

Strange thing is, your suggestion for deriving vol of vol from a time-series is what I did some time ago. However, this was before I came the term "vol of vol".

I am familiar with mean reversion but kappa is a new one. I've always imagined that the further implied moves from the mean, the greater the pull is (similar to "pull-to-par"?) to bring it back - almost like an elastic band affect.

Can kappa be quantified or is it just intuitive, eg kappa will be greater for implied the further the implied is from the mean, that is, the speed of the return will be that more rapid (is that a tautology)?

"Strikes" and "liabilities". So what's being referred to here?

Thanks, once again.

Grant.

Here is a good paper about mean reversion and how to measure it...http://www.actuaries.org.uk/files/pdf/proceedings/fib2004/Exley_Mehta_Smith.pdf

The volatility of the implied vol time series is vol of vol!!! The implied vol is a stochastic process (mean reversion process, see paper) and hence will have a variance (volatility). Even the vol of vol is a stochastic process so one could, if they really wanted to, could find the vol of vol of vol.....etc etc etc (http://www.ingber.com/markets99_vol.pdf)

i.e your underlying process dS/ S = mu.dt + sig(t).sqrt[dt].dW1

where sig = kappa*[sig-m].dt + volOfVol*dW2 ..... i.e your vol is now stochastic

where volOfVol if your vol of vol

grantx · Dec 7, 2006

Robert,

Thank you for the references. Heavy stuff. I'll certainly need to get my brain in gear here. Doubtless, will have to come back for clarification.

Thanks, Robert.

Grant.

Derman, Implied and local volatility, etc

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